While this money cannot help reduce unemployment and can only slow the recovery ;letting it expire will not have a noticeable positive effect in the short term. There is no way to measure the effect so politically there is no short term benefit to letting it expire, only a drawback. In the long term Obama gets the blame we assume.
Now let's go to Democrats so called economic theory : That “ Unemployment compensation creates jobs because the money is spent immediately creating the needed demand for new workers” . Given that politically convenient theory Democrats accuse Republicans of trying to hurt the recovery by blocking this key ‘jobs’ bill. (The Democrat jobs bill is paying people to not get a job.)
Democrats plans, keeping the country unemployed works much better if you have a Republican President to blame. Oh, do they miss Bush ever.
Timing is everything. Get the unemployment benefits extended beyond the November midterms and deal with it again during the lame duck session. Obama would rather continue the payments than have voters even angrier that they have no jobs.